\chapter{Application: Portfolio Choice}
%Investors with high expecation are more sensitive to the realization of result, since they have larger chances of losses. Since people are loss averse, therefore, providing more fluctuations in utility. 
%However, whether higher sensitivity is good or bad is determined by the real distribution of the underlying asset. For sophisticated ones, who excluded the effects from anticipatory utility, still could not avoid their sensitivity to the results in loss region. People are always sensitive to the lower ranking returns than to the higher ranking returns, determined by the loss aversion. Therefore, 
%They have in general stronger feelings of results. when the returns are good, then, stronger feelings are better, therefore, take more risks. Otherwise, take less risks. 
%Great!! Finally got it!

In this chapter, I explore some implications of my model in investment problem--a biased investor with concave utility function choosing optimal portfolio of a combination between one risky and one risk-free assets to maximized her utility. Continue with the assumption in last chapter, I assume there exist two type of investors: the naive type who maximized the expected return and the sophisticated type who maximized the total utility including both expectation and future possible gain-loss. We conclude that a naive type and a sophisticated type can adopt opposite investment strategy even though they form their subjective beliefs through similar cognitive process. We further derive the equilibrium in a market with investors holding homogeneous beliefs and gives implications on pricing of assets. 
\section{Choice between one risk-free asset and one risky asset}\label{riskfree}

\begin{ass}
The return of an asset has continuous distribution function $f(\cdot)$ defined over $(-\infty,+\infty)$. The investor holds subjective belief $g(\cdot)$ of the asset's return, which is also a continuous $p.d.f.$ on $(-\infty,+\infty)$. 
\end{ass}

There are two assets in the market: a free risk-free asset with return $R_f$ and infinite supply; and a risky asset with gross return $R_R=R_f+R_m=R_f+E_{f(\cdot)}(R)$, where $R_m$ is the gross excess return, $R=R_r-\pi$ is the realized excess return $R_r$ adjusted by the equilibrium price $\pi$. The consumption utility takes the same assumption as before which is $u'(x)>0, u''(x)\leq0$. There are two periods. In period 1, the agent simultaneously forms her subjective optimal beliefs $g(R)$ about payoff of the risky asset and allocates her unit endowment between these two assets. In period 2, the payoffs of the assets are realized. 

Continue with previous separation on "naive" and "sophisticated" agents, I assume that the naive type has different goal functions in choosing optimal subjective beliefs and optimal $\alpha$. The choice of portfolio are rational choice based on their biased beliefs. The sophisticated type chooses optimal subjective beliefs and optimal $\alpha$ allocated to the risky assets simultaneously. 

Specifically, at T=1, we assume that for any given optimal beliefs $g(R)$, an naive agent chooses her portfolio share, $\alpha^{BS}$, to invest in the risky asset to maximize the expected return: 
\begin{equation*}
W=\underset{\alpha}{Max}\int_{-\infty}^{+\infty}g(R)u(R_{f}+\alpha R)\mathrm{d}R. 
\end{equation*}

Given optimal choice of $\alpha^{BS}$, the naive agent chooses her subjective beliefs $g(\cdot)$ to solve the the following problem: 
\begin{equation*}
\underset{g(R)}{argmax}\: E_{g(R)}u(R_{f}+\alpha^{BS}R)+E_{f(R)}\mu[u(R_{f}+\alpha^{BS}R)-E_{g(R)}u(R_{f}+\alpha^{BS}R)] 
\end{equation*}
Instead, we assume that a sophisticated agent simultaneously chooses her optimal portfolio share $\alpha$ and optimal beliefs $g(R)$ to solve: 
\begin{equation*}
\underset{g(R),\: \alpha}{argmax}\: E_{g(R)}u(R_{f}+\alpha R)+E_{f(R)}\mu[u(R_{f}+\alpha R)-E_{g(R)}u(R_{f}+\alpha R)] 
\end{equation*}

Brunermeier and Parker(2005) built a discrete model in which an agent has optimal beliefs determined by \\
\begin{equation*}
\underset{\{q_{s}\}}{argmax}\:\overset{S}{\underset{s=1}{\Sigma}}q_{s}u(R_{f}+\alpha^{BS}R_{s})+\overset{S}{\underset{s=1}{\Sigma}}p_{s}u(R_{f}+\alpha^{BS}R),
\end{equation*}
where $\alpha^{BS}$ is the solution to $\underset{\alpha}{max}\:\overset{S}{\underset{s=1}{\Sigma}}q_{s}u(R_{f}+\alpha R_{s})$. 
This is a special case of the naive agent when $\lambda=1$ and $\eta=\dfrac{1}{2}$. 

In their paper, Brunermeier and Parker concluded that optimal beliefs holders always trading more aggressively than rational agent since the cost from distorted portfolio choice is second order while the benefits from a higher anticipation is first order. 

The following proposition indicates that reference-dependent loss averse investors no longer always trade aggressively. Even though the cost of distorted portfolio is still second order, however, biased beliefs are not only first order generator of benefits from higher anticipation, but also give first punishment from the "reference effect". Therefore, biased beliefs holders can have aggressive or conservative trading strategy depending on their optimistic and pessimistic attitudes. 

\subsection{Naive Agent}

\begin{prop}{(Risk taking due to optimism and pessimism: Naive Case):}\label{riskfreenaive}\\
An optimistic investor with low $P^*$ invests more aggressively than an agent with rational beliefs or in the opposite direction; a pessimistic investor with high $P^*$ invests in the same direction as the rational investor but more conservatively: \\
if $E(R)>0$, then $\alpha^{RE}>0$,\\
and $\alpha^{OP}>\alpha^{RE}>0$ or $\alpha^{OP}<0<\alpha^{RE}$,\\
$0<\alpha^{PE}<\alpha^{RE}$;\\
if $E(R)<0$, then $\alpha^{RE}<0$, \\
and $\alpha^{OP}<\alpha^{RE}<0$ or $\alpha^{OP}>0>\alpha^{RE}$,\\
$\alpha^{RE}<\alpha^{PE}<0$. 
\end{prop}
Similar to Burnnermeier and Parker(2005), the optimal $\alpha^{BS}$ under biased beliefs are always different from the $\alpha^{RE}$ since biased beliefs ensures higher total welfare. Optimistic investors in our model trade in the same direction but more aggressively than a rational agent since they overestimate the chance of success investments or they will take a opposite position to the rational strategy. Same reason as BP states: opposite trading happens when the asset is skewed enough in the opposite direction of the mean payoff. 

Different from BP's conclusion, investors no longer always hold "optimistic" beliefs or trade aggressively. Instead, a pessimistic investor characterized by a high $P^*$ invests in the same direction but more conservative than their rational counterpart.\footnote{"Optimistic" and "Pessimistic" have different meaning from Brunermeier and Parker(2005). BP's "Pessimistic" investors assign higher probabilities to negative returns while short. Pessimistic defined in this model means "conservative": pessimistic investors assign lower probabilities to negative returns while still short.} Optimal expectation holders in BP's model behaves exactly the same as the optimistic investors in my model. In fact, BP's model describes the extreme case of my model in which $\lambda=1$ and $\eta=1$. $\lambda=1$ represents an optimistic investor with no loss aversion and has $P^*=0$. Investors in BP's model always trade aggressively because the additional anticipation is pure generator of felicity. Since the only punishment comes from the distorted portfolio choice, both up- or down- biases reduce the realized utility. However, up-biases dominate down-biases because of the extra felicity from good anticipation. Therefore, all investors are more optimistic than rational and trade aggressively. \footnote{Mathematically, BP's model requires $\alpha^{BS}>\alpha^{RE}>0$ because their first order maximization problem requires this condition. The envelope condition is $(u_{s''}-u_{s'})\Delta\hat{\pi}+\eta\Sigma p_{s}u'(R_{f}+\alpha^{BS}R_{s})R_{s}\mathrm{\Delta\alpha}=0$. When $\alpha^{RE}>\alpha^{BS}>0$, $\Sigma p_{s}u'(R_{f}+\alpha^{BS}R_{s})R_{s}>0$, then biases in beliefs serves as a utility pump thus gives no limits to biases. }
%\end
%$\alpha^{RE}>\alpha^{BS}>0$, then biases in beliefs serves as a utility pump thus gives no limits to biases. This condition, however, is externalized to their model and therefore logically when the second order material cost is insufficient, people can boost their bias in beliefs to any level they prefer.} 


 %This conclusion can be seen from the envelope condition: 
%\begin{equation*}
%(u_{s''}-u_{s'})[1+\eta(1-\lambda)p_{\tilde{s}}]+\eta[\Sigma p_{s}u'(R_{f}+\alpha^{BS}R_{s})R_{s}+(\lambda-1)\underset{Loss}{\Sigma}p_{s}u'(R_{f}+\alpha^{BS}R_{s})R_{s}]\mathrm{\Delta\alpha}=0 
%\end{equation*}
%where $p_{\tilde{s}}$ is the marginal state transferred from the gain to loss region due to the small changes in beliefs. 
%When $\lambda=1$, we have 
%\begin{equation*}
%(u_{s''}-u_{s'})\Delta\hat{\pi}+\eta\Sigma p_{s}u'(R_{f}+\alpha^{BS}R_{s})R_{s}\mathrm{\Delta\alpha}=0 
%\end{equation*}

%which is exactly the envelope condition in BP they used as the key proof to $\alpha^{BS}>\alpha^{RE}>0$. 

%The reason that BP's model requires $\alpha^{BS}>\alpha^{RE}>0$ is because if $\alpha^{RE}>\alpha^{BS}>0$, then biases in beliefs serves as a utility pump thus gives no limits to biases. This condition, however, is externalized to their model and therefore logically when the second order material cost is insufficient, people can boost their bias in beliefs to any level they prefer. The inclusion of anticipation into the realization period internalized the boundary on beliefs while gives source to optimal bias in beliefs. To be specific, my model allows investors to be conservative because of the cooperation of subjective expectation and loss aversion ensures that the marginal cost from changing wealth allocation comes not only directly from the increasing or decreasing in marginal utility, but also provides extra utility from the kink point between gain and loss. 
%It is obviously that there always exist an optimal $\alpha^{OP}$ such that the envelope condition is satisfied. For the pessimistic agent, however, though the direct marginal cost is positive, the extra cost from loss will come to balance. Since a pessimistic agent characterized by a high $P^*$ usually has a high $\lambda$ and a low expectation and these two factors will finally lead to a negative marginal cost from moving up the value of $\alpha$. 

Furthermore, even though we define "optimistic" and "pessimistic" in terms of low or high $P^*$, however, by saying that $P^*$ is low, we in fact comparing $P^*$ with real chance of gain which is determined by the distribution of the returns. An agent with a low $P^*$ tend to overestimate the chance of good returns no matter they are in the short or long position, but the same person can exhibit opposite attitudes in different positions of assets. For example, in face of a negative skewed asset with $E(R)<0$. An investor choose the short position will be pessimistic since the long tail on the left gives few chances of gains; instead, the same investor if enters into the long position can be optimistic, since the short tail on the right give high chances of gain. The choice of attitude depends on both the investment position as well as the skewness of the asset. 

The construction of optimization problem for an naive agent indicates that the agent chooses $\alpha$ after she makes her choice of beliefs in a non-strict way. This is because we assume in making portfolio choice, she takes beliefs as given and fail to recognize her biases. Therefore, optimistic and pessimistic attitudes directly affect the value of $\alpha$. Intuitively, an optimistic agent prefers investing more aggressively since higher proportion of the risky asset in portfolio gives more promising returns than it actually does while an pessimistic investor who over-estimate the bad side of the risky assets would certainly cut the proportion of a risky one. We arrive at this conclusion mainly because the choice of $\alpha$ happens "after" the choice of beliefs and therefore deviation in $\alpha$ from rational level, like BP said, only put on second order cost and changes in beliefs gives first order increase in anticipation and dominates the total effects. However, people may exhibit more complexity than the described naive type, in which case, when choosing optimal portfolio, they may also realize their biases in beliefs and make their investment decision accordingly. In the following part, we discuss the behaviour from a sophisticated agent in our portfolio choice problem. 

%Moreover, from proposition \ref{riskfreenaive},  we see that an optimistic investors behaves just like the agents in the model of Brunnermeier and Parker(2005). The intuition is similar as well. A small up-bias in beliefs lead to a first order gain in anticipatory utility which dominates the change in total utility. Therefore, for $\alpha^{RE}>0$, an optimistic investor, when decides to long the risky asset, will over-estimate the probabilities of high payoffs and take excess risks, or have too much confidence in the opposite direction and short. However, different from their model, the bias in beliefs no longer only provides first order gains, but will also serves as the source of loss. An bias in beliefs lead to a first order loss from the second part of total utility since the anticipation, as the reference point increases. Therefore, even though an optimistic investor who weight heavy on current anticipatory utility still behaves the same as in Brunnermeier and Parker's model, the fundamental logic is different. Their aggression now mainly due to 

%Notice that there is a special case of Proposition \ref{riskfree} when the excess return under the true probability distribution $f(\cdot)$ is 0, that is $R_m=E_{f(\cdot)}(R)=0$. The proportion of endowment invested in the risky asset $\alpha$ is 0 if beliefs are unbiased since investors are risk averse. However, an optimistic or pessimistic investor with biased beliefs are still willing to long or short the risky asset with positive or negative $\alpha$. In particular, even though investors' preferences are not identical to each other, they tend to long the asset with moderate negative skewness and with extremely high positive skewness (as both have relatively high cumulative probability on the right tail) while short the asset with moderate positive skewness and with extremely low negative skewness. 

\subsection{Sophisticated Agent}

\begin{prop}{(Risk taking due to optimism and pessimism: Sophisticated Case):}\label{riskfreeso}
\item[i] 

Biased investors invest more aggressive than rational investors if the average return conditional on loss is good enough. In other cases, biased investors invest either more conservative or in the opposite directions to the rational strategy. To be more specific, 

$\alpha^{BS}>\alpha^{RE}>0$ ($\alpha^{BS}<\alpha^{RE}<0$) if\\ $E(R)>(<)0$ and $\underset{-BS}{\int}f(R)u'(R_{f}+\alpha_{RE}R)R\mathrm{dR}>(<)0 $\\
$\alpha^{BS}<\alpha^{RE}$ ($\alpha^{BS}>\alpha^{RE}$) if\\ $E(R)>(<)0$ and $\underset{-BS}{\int}f(R)u'(R_{f}+\alpha_{RE}R)R\mathrm{dR}<(>)0 $, \\
where $\alpha^{BS}$ and $\alpha^{RE}$ are the optimal allocation of wealth on risky assets and "-BS" indicates the region of loss under biased beliefs taking $\alpha_{RE}$ as given.

\item[ii]
For $R_{CE}>0$, $\alpha^{BS}$ decreases in $P^*$;\\ 
For $R_{CE}<0$, $\alpha^{BS}$ increases in $P^*$, \\
where $E_{g(\cdot)}u(R_f+\alpha R)=u(R_f+\alpha R_{CE})$ and $R_f+R_{CE}$ is the certainty equivalent. 
\end{prop}
%Optimistic agents invest more aggressively than pessimistic agents if the return of the assets is perceived as good on average;
%Pessimistic agents invest more aggressively than optimistic agents if the return of the asset is perceived as bad on average. 
%Formally, suppose $E_{g(\cdot)}u(R_f+\alpha R)=u(R_f+\alpha R_{CE})$, where $R_f+R_{CE}$ is the certainty equivalent and $E_{g(\cdot)}u(R_f+\alpha R)$ is the expectation under optimal subjective beliefs. 
%For $R_{CE}\geq0$, we have, 
%$$
%0<\alpha^{PE}<\alpha^{OP} or \alpha^{PE}<\alpha^{OP}<0 
%$$
%For $R_{CE}<0$, we have, 
%$$
%\alpha^{OP}<\alpha^{PE}<0 or \alpha^{PE}>\alpha^{OP}>0
%$$
%where $\alpha^{OP}$ and  $\alpha^{PE}$ are the proportion of wealth allocated to risky asset with optimal optimistic and pessimistic beliefs respectively. 
%\end{prop}

%For $\alpha^{RE}\geq0(<0)$ and $R_{CE}\geq0$, we have, 
%$$
%\alpha^{PE}<\alpha^{RE}<\alpha^{OP}  (\alpha^{OP}<\alpha^{RE}<\alpha^{PE});
%$$
%For $\alpha^{RE}\geq0(<0)$ and $R_{CE}<0$, we have, 
%\alpha^{OP}<\alpha^{RE}<\alpha^{PE} (\alpha^{PE}<\alpha^{RE}<\alpha^{OP});where $\alpha^{RE}$ is the optimal wealth proportion allocated to the risky asset under rational unbiased beliefs, $\alpha^{OP}$ and  $\alpha^{PE}$ are the proportion to risky asset with optimal optimistic and pessimistic beliefs respectively. \end{prop}

Unlike naive agents, proposition \ref{riskfreeso} indicates that the behaviour of the sophisticated agents are more complicated. Like our analysis in Chapter 4, a sophisticated agent considers both anticipation and the "reference effect" in making her portfolio choice. At optimal beliefs, these two effects cancel out each other, leaving with rational-like expected utility. However, loss averse investors are different from rational investors since they have different sensitivities towards future payoffs, depending on their choice of reference level.  Basically, optimism, by introducing in more losses, also makes people more sensitive towards outcomes while pessimism, by reducing the fear of loss, makes people less sensitive to outcomes. 

The first part of Proposition\ref{riskfreeso} describe the difference in portfolio choice decision between a sophisticated biased agent and a rational agent. The difference between a rational agent and the sophisticated agent lies in whether they assign additional weight to the loss region. The intuition is simple: compared with rational agents, loss averse agents care more about lower rank outcomes, therefore, when low-rank returns are good on average, loss averse agents are happier than the rational ones and can tolerate more risks. On the other hand, bad returns on the lower rank outcomes are more painful for loss averse agents thus lead to conservatism or opposite trading strategy.  

The second part of Proposition \ref{riskfreeso} compares the investment strategies among sophisticated agents with different optimistic and pessimistic attitudes. To get more clear intuition, consider the following lemma directly derived from Proposition \ref{riskfreeso} part 2. 

\begin{lemma}\label{riskfreesolemma}
Optimistic agents invest more aggressively than pessimistic agents if the return of the assets is perceived as good on average;
Pessimistic agents invest more aggressively than optimistic agents if the return of the asset is perceived as bad on average. 
For $\alpha^{BS}>(<)0$ and $R_{CE}>0$, we have $\alpha^{OP}>\alpha^{PE}>0$ ($\alpha^{PE}<\alpha^{OP}<0$);
for $\alpha^{BS}>(<)0$ and $R_{CE}<0$, we have $\alpha^{PE}>\alpha^{OP}>0$ ($\alpha^{OP}<\alpha^{PE}<0$);
\end{lemma}

To better understand the intuition, consider the following example. Suppose there is an investor in the long position of a security and suppose the security gives good returns on average in the future which is known by the investor. Even though a pessimistic sophisticated investor known that her low anticipation can avoid losses in the future, however, being pessimistic also makes her numb since she has less fear for losses and thus less care for the outcomes. We say that she is numb because losses are source of strong feelings compared to gains. By avoiding losses, she also avoid intensive feelings. For this reason, good future prospects provides less felicity. Instead, an optimistic investor with understanding of large chance of future loss cares more about the security due to her fear of loss. More attention to the security gives stronger feeling on returns. Therefore, when the returns are in generally good, optimistic investors enjoys more felicity than a pessimistic one and an optimistic investor is willing to hold more risky asset in her portfolio. 

On the contrary, when the security gives a bad return on average, being pessimistic reduces the fear for loss and makes people less sensitive about future payments. Therefore, when the asset is a bad one, pessimistic and numbness make the bad outcomes more tolerable and pessimistic agents can bear extra risks than optimistic ones. 

Compared with the bad return situation, when the asset has promising good returns, even though staying numb can make pessimistic ones bear more risks, however there are little chance to feel bad since the asset is good. Therefore, optimism is more beneficial since it intensifies the happiness from good outcomes and enables optimistic investors to bear more risks than pessimistic ones.

%Changing beliefs would change the value of expectation, therefore, enclose more or less return levels into the "gain" region. However, the marginal utility by doing so is $u'(R_f+R)R$, depends on whether the return is positive or negative. An increase in $\alpha$ leads to a second order decrease in utility that is going to be realized as marginal utility is decreasing: $u"(\cdot)\leq0$. Optimal beliefs must further change to compensate the decrease in utility. When the marginal utility is positive, then further up-bias is beneficial since it gives a higher first-order utility as $u'(\cdot)\geq0$. Therefore, if an investor is optimistic at her optimal beliefs, then, bring back her beliefs to the rational level would give her a first order decrease in anticipatory utility. Therefore, the investor must choose a lower $\alpha$ to increase the marginal utility. Instead, when the marginal certainty equivalent is negative, bring down the beliefs would lead to a first order increase in marginal utility. The investor wants to hold more risky assets to increase the total utility. Similar analysis for pessimistic investors. Compared to naive agents, optimistic investors are no longer simply holding more risky assets as they over evaluate its the return and pessimistic investors are no longer just shorting the risky assets, instead, both optimistic and pessimistic investors can hold more or less proportion of the risky assets than their rational counterpart. 

\section{Equilibrium}
In this section, we place the portfolio choice problem into an exchange economy with identical agents holding homogeneous beliefs. We study the security pricing in the market with only two assets, a free risk-free asset with $R_f=0$ and price $\pi_f=0$ and a risky asset with stochastic excess return $R$ and price $\pi_r$. The distribution of $R$ with p.d.f $f(\cdot)$ and c.d.f. $F(\cdot)$  is public known by the investors. We assume the short-sale constraint binds, therefore, the proportion of wealth allocated to the risky asset $\alpha$ satisfy $0 \leq \alpha \leq 1$. We further simplify the assumption by making utility function take the linear format: $u(x)=x$. 
The simplest candidate equilibrium is a homogeneous holdings equilibrium: an equilibrium in which all investors hold the same portfolio. However, for both naive and sophisticated investors, as we proved latter, the optimal portfolio under the linear utility function assumption is either holding all risky asset or holding all risk-free asset. Therefore, the equilibrium which is constructed by such kind of portfolio choice must ensure the utility from holding only the risky asset is no different from the utility from holding the risk-free asset. 

\paragraph{Naive}
Previous problem of an naive agent under the new assumption becomes: 
$$
g^{*}(R)=\underset{g(R)}{argmax}\underset{-\infty}{\overset{+\infty}{\int}}g(R)(R_{f}+\alpha^{*}R)\mathrm{dR}+\underset{-\infty}{\overset{+\infty}{\int}}f(R)[R_{f}+\alpha^{*}R-\underset{-\infty}{\overset{+\infty}{\int}}g(r)(R_{f}+\alpha^{*}r)\mathrm{dr}]\mathrm{dR}
$$ 


\begin{equation*}
\alpha^{*}=\underset{\alpha\in[0,1]}{argmax}\underset{-\infty}{\overset{+\infty}{\int}}g^{*}(R)(R_{f}+\alpha R)\mathrm{dR}, 
\end{equation*}

where $\pi^*_N$ is the equilibrium price in the market with naive agents.
It is obvious from the optimization problem that for $R_f=0$, we have\\
$\alpha^*=1$ if $E_{g^{*}}R\geq 0$\\
$\alpha^*=0$ if $E_{g^{*}}R\leq 0$\\
In the equilibrium, we have, the price of the risky asset today $\pi_{N}$ satisfies,\\
$R_{f}-0=\eta\cdot E_{g^{*}}R-\pi_{N}$. 
The equilibrium price is therefore, 
$$
\pi^{*}_{N}=\eta \cdot E_{g^{*}}R 
$$
Instead, in a market rational investors who maximized expected returns holding unbiased beliefs, the equilibrium price of the risky asset is determined by
$\pi^{*}_{RE}=\eta\cdot E_{f^{*}}R$

\paragraph{Sophisticated}
Now, consider the security pricing if investors are sophisticated. The optimization problem is, 
$$
\underset{g(R),\alpha}{max}\underset{-\infty}{\overset{+\infty}{\int}}g(R)[R_{f}+\alpha R]\mathrm{dR}+\underset{-\infty}{\overset{+\infty}{\int}}f(R)\{R_{f}+\alpha R-\underset{-\infty}{\overset{+\infty}{\int}}g(r)[R_{f}+\alpha r\mathrm{dr}\}\mathrm{dR}
$$

From the first order condition with respect to $\alpha$, we have, \\
$\alpha=1$ if $\underset{-\infty}{\overset{+\infty}{\int}}f(R)R\mathrm{dR}+(\lambda-1)\int_{-\infty}^{E_{g^{*}}}f(R)R\mathrm{dR}-\pi_{S}>R_f-0$\\
$\alpha=0$ if $\underset{-\infty}{\overset{+\infty}{\int}}f(R)R\mathrm{dR}+(\lambda-1)\int_{-\infty}^{E_{g^{*}}}f(R)R\mathrm{dR}-\pi_{S}<R_f-0$\\

In equilibrium, we have, for $\alpha^{*}=1$
\begin{equation*}
\underset{-\infty}{\overset{+\infty}{\int}}f(R)R\mathrm{dR}+(\lambda-1)\int_{-\infty}^{E_{g^{*}}}f(R)R\mathrm{dR}-\pi^{*}_{S}=R_f=0
\end{equation*}
Therefore, the equilibrium price for a market with identical sophisticated agents is \\
\begin{equation*}
\pi_{S}^{*}=\eta E_f(R)+\eta(\lambda-1)\int_{-\infty}^{E_{g^{*}}}f(R)R\mathrm{dR}. 
\end{equation*}
%Instead, if investors are rational holding unbiased beliefs, we have
%\begin{equation*}
%\pi_{SR}^{*}=E_{f}(R)+\eta(\lambda-1)\int_{-\infty}^{E_{f}}f(R)R\mathrm{dR}-\eta(\lambda-1)E_{f}(R)\cdot\int_{-\infty}^{E_{f}}f(R)\mathrm{dR}
%\end{equation*}. 

To gain further intuition on the equilibrium, consider the a normal distributed asset with $R\sim N(1, 1)$.  $\lambda$ is set at $2.25$ as suggested by experiment evidence. Variation in $\eta$ changes $P^*$ from $0$ to $1$.\footnote{$lambda$ and $\eta$ can change simultaneously, however, for calibration purpose, we set lambda at 2.25.} The result is shown in Figure 1 and Figure 2. 

%\begin{figure}
%\includegraphics{EP.jpg}
%\end{figure}

The solid lines in Figure 1 and Figure 2 plot the rational price as a function of  $P^{*}$. Since $P^{*}$ is increasing in $\eta$, the equilibrium price increases to rational expectation as the discount rate $\eta$ goes to 1. The dash line in Figure 1 plots the equilibrium price against $P^{*}$ in a market with naive investors.  Consistent with the prediction of Proposition \ref{riskfreenaive} and since the short-sale constraint binds, the equilibrium price of the risky asset decreases as investors become more pessimistic. To be more specific, the "naive" price is higher than rational level when the market is dominated by optimistic investors because the up-biased beliefs urge the investors to hold more risky asset. Instead, the "naive" price falls below rational level when investors are pessimistic. Pessimistic investors require higher equity premium to compensate for their stronger aversion to loss.

%To gain further intuition on the equilibrium, consider the following example with previous discussed assets: a normal distributed asset with $R\sim N(1, 1)$ and a positive skewed asset with $R \sim Gamma(4, 0.5)$. Under these assumptions, these two assets share the same expectation and variance. Further more, $\lambda$ is set at $2.25$ as suggested by experiment evidence. Variation in $\eta$ changes $P^*$ from $0$ to $1$. The result is shown in Figure X and X. 

The dash line in Figure 2 plots the "sophisticated" equilibrium price as a function of $P^*$. Compared with the "naive" market, the sophisticated price exhibit U-shape other than strictly decreasing in $P^*$. The equilibrium price is upper-bounded by the rational expectation implying our conclusion that sophisticated agents act similar to rational agents. Further more, consistent with prediction of Proposition\ref{riskfreeso}, with short sale constraint binds, both the optimistic and the "very" pessimistic agents price the asset higher than moderate pessimistic agent. With short-sale constraint binds, optimistic agents for this symmetrically distributed asset are those with $P^{*}<0.5$ while pessimistic agents have $P^{*}>0.5$. Since $\mu=1$, therefore, $R_{CE}^{OP}=E_g(R)>0$ and utility increases as agents become more optimistic, so does the price. Since the average return captured by the optimistic agents is always greater than 0, therefore, the asset is a good one and higher reference point intensify the good feeling for good returns. 

On the contrary, for "extremely" pessimistic agents (in this case, $P^{*}>0.63$), they have $R_{CE}^{PE}=E_g(R)<0$ and the desirability of the risky asset is also increasing as agents turn more pessimistic. Since the return of the asset is not good enough for those very pessimistic agents to bias up, they subjectively capture a bad average return. Further pessimism reduces sensitivity toward bad returns and increase tolerance of risks. 

Finally, for "moderately" pessimistic agents who have $0.5<P^{*}<0.63$ in this example, correctly captures the goodness of the asset, that is they have $R_{CE}^{PE}=E_g(R)>0$. They are pessimistic since the asset fails to give sufficient chance of good returns to persuade them into optimistic. By being more pessimistic, they reduces the deserved happiness from good outcomes instead of reducing the painfulness from bad outcomes. Therefore, they are the most "risk-averse" ones among these three groups. 

Furthermore, rational price is lower than "sophisticated" price when investors are optimistic and exceeds the latter as investors become more pessimistic. Consistent with the first part of Proposition \ref{riskfreeso}, since $\mu>0$, therefore, the average return on the loss region increases from negative to positive as confidence increases. Consequently, intensive feeling on bad returns push down the sophisticated price when investors are pessimistic and boost up it  when the market is optimistic. 


%In Figure 1, the blue line and the red line plot the equilibrium price for a normal distributed assets and the gamma distributed assets in the market with naive agents with different confidence level indicated by $P^*$. The black line is the rational pricing in the market with non-biased naive agents. Rational pricing lies on the rational expectation which is equals to 2. Equilibrium price increases as investors become more optimistic, decreases as investors become more more pessimistic. The skewed asset, since having higher probability weights on extreme values, is more valued than the non-skewed asset in a market with extreme optimistic or pessimistic investors which is consistent with proposition \ref{twoindepgen}. This result is consistent with our predictions. With short-sale constraint binds, Proposition \ref{riskfreenaive} states that the willingness of holding the risky asset decreases as people become more pessimistic. Therefore, in equilibrium, market price the risky asset higher(lower) than the rational equilibrium price when naive optimistic(pessimistic) investors are trading in the market. 

%Further more, in Figure2, the blue and red solid lines plot the equilibrium price for our non-skewed and skewed asset in the market with sophisticated agents holding biased beliefs while the dashed lines are pricing of securities for sophisticated but unbiased agents. Price in the market with sophisticated but rational investors is decreasing as pessimism indicated by $P^*$ increasing. No surprising, the result is due to a higher weight on future gain-loss utility. ???Prices under biased beliefs are always higher than unbiased beliefs which is consistent with prediction of Proposition \ref{}. For optimistic investors, $R_{CE}>0$, they are willing to possess more risky asset than a rational investor will do; for pessimistic investors, $R_{CE}<0$, they are also willing to possess more risky asset.???? Therefore, equilibrium price is always higher than the rational price. Notice that comparing with the price under naive rational beliefs(also the rational defined by classical economic theory), sophisticated investors always gives a lower price. This result can be seen directly from Proposition\ref{lotterys}. Since $E_f(Z)=0$, utility from a sophisticated agent with biased beliefs is always below the rational expected utility. Equilibrium price exhibit u-shape in $P^*$, indicating that the market with moderate level of confidence requires higher risk-premium. 